Carmichael numbers

A Carmichael number is a pseudoprime for every possible base b:
that is, for every b coprime to N. It was recently shown that
there infinitely many of these numbers too.
We have computed all Carmichael numbers up to 1021 and various
other lists: see tables and statistics. Details
of the computation are in a number of papers.

Carmichael numbers up to various limits

Carmichael numbers up to 1015

There are 105212 Carmichael numbers up to 1015: we describe
the calculations. The numbers were generated by a back-tracking search for
possible prime factorisations, and the computations checked by
searching selected ranges of integers directly using a sieving technique,
together with a ``large prime variation''.

Carmichael numbers up to 1016

We extend our previous computations to show that there are
246683
Carmichael numbers up to 1016. As before,
the numbers were generated by a back-tracking search for
possible prime factorisations together with a
``large prime variation''. We present further statistics on
the distribution of Carmichael numbers..

On using Carmichael numbers for public key encryption systems

We show that the inadvertent use of a Carmichael number instead of a prime
factor in the modulus of an RSA cryptosystem is likely to make the system
fatally vulnerable, but that such numbers may be detected.

Pseudoprimes

A Fermat pseudoprime base b is a composite number N which
satisfies the Fermat condition bN-1
congruent to 1 modulo N. In the common case b = 2 we often
talk simply of a pseudoprime. The first few examples are 341 = 11.31,
561 = 3.11.17, ... . It is not too hard to show that there are infinitely
many: if n is a pseudoprime, so is 2n-1.

Pseudoprimes up to 1013

There are 38975 Fermat pseudoprimes (base 2) up to 1011, 101629 up
to 1012 and 264239 up to 1013:
we describe the calculations and give some statistics.
The numbers were generated by a variety of strategies, the
most important being a back-tracking search for possible prime
factorisations, and the computations checked by a sieving technique..